Signal and noise of Fourier reconstructed fMRI data

Abstract

In magnetic resonance imaging, complex-valued measurements are acquired in time corresponding to spatial frequency measurements in space generally placed on a Cartesian rectangular grid. These complex-valued measurements are transformed into a measured complexvalued image by an image reconstruction method. The most common image reconstruction method is the inverse Fourier transform. It is known that image voxels are spatially correlated. A property of the inverse Fourier transformation is that uncorrelated spatial frequency measurements yield spatially uncorrelated voxel measurements and vice versa. Spatially correlated voxel measurements result from correlated spatial frequency measurements. This paper describes the resulting correlation structure between voxel measurements when inverse Fourier reconstructing correlated spatial frequency measurements. A real-valued representation for the complex-valued measurements is introduced along with an associated multivariate normal distribution. One potential application of this methodology is that there may be a correlation structure introduced by the measurement process or adjustments made to the spatial frequencies. This would produce spatially correlated voxel measurements after inverse Fourier transform reconstruction that have artificially inflated spatial correlation. One implication of these results is that one source of spatial correlation between voxels termed connectivity may be attributed to correlated spatial frequencies. The true voxel connectivity may be less than previously thought. This methodology could be utilized to characterize noise correlation in its original form and adjust for it. The exact statistical relationship between spatial frequency measurements and voxel measurements has now been established.

Incorporating relaxivities to more accurately reconstruct MR images

“…When the complex-valued Fourier reconstruction is described through a real-valued isomorphism [17], a vector of the reconstructed image, y , can be written as the product of a FR operator, Ω, with a vector of the observed k -space observation, s , by…”

“…In order to relate the signal and noise characteristics of k -space measurements to reconstructed voxel measurements, the complex-valued matrix application of the inverse Fourier transformation was described through a real-valued isomorphism by Rowe et al [17]. Representing the Fourier reconstruction as a single matrix operator formed the basis for the study in [18] where a mathematical framework, AMMUST- k (A Mathematical Model for Understanding the STatistical effects of k -space preprocessing), was developed to represent various spatial processing operations performed on acquired spatial frequencies in terms of real-valued linear isomorphisms.…”

“…Although T 1 on tissue relaxation time provides valuable quantitative information characteristics, the T 1 recovery term is typically neglected by assuming a long repetition time. In this study, the linear framework presented in the work of Rowe et al, 2007, and of Nencka et al, 2009 is extended to develop a Fourier reconstruction operation in terms of a real-valued isomorphism that incorporates the effects of T 2 *, Δ B , and T 1 . This framework provides a way to precisely quantify the statistical properties of the corrected image-space data by offering a linear relationship between the observed frequency space measurements and reconstructed corrected image-space measurements.…”

Shape Preserving Fitting Model for Affective Curves Extraction: An Affective Computing Method on fMRI Dataset

“…Among many of those tools, fuzzy set was considered to be an effective tool, which was introduced by L.A. Zadeh in 1965, it was different from the classic set collection system and used a membership function to react the degree of reaction thing belongs, fuzzy inference system was applied in affective computing [14]and medical fields [15]. In the fMRI data preprocessing, principal component analysis (ICA) [16], Fourier transform [17], and spectrum calculation [18], are very common methods. Results for image segmentation means of computer image processing, such as Gaussian filtering, it also included some De-noising practices.…”

Quantifying the Statistical Impact of GRAPPA in fcMRI Data With a Real-Valued Isomorphism

“…2(a). To reconstruct a vector of spatial frequencies into voxel values, the real-valued matrix representation of the complex-valued IFT, Ω, was derived by Rowe et al 19 At this stage, one can either reconstruct each coil image, using a Kronecker product, ( ⊗ Ω), and apply a combination matrix, , to perform the combination in the image domain = ( ⊗ Ω) full…”